A vibrating mass is found to be oscillating with an amplitude that is too large. To reduce this amplitude, an auxiliary system is added as shown in Figure 6.79. This problem generally occurs when the forcing frequency is too close to the natural frequency of the primary system.

The auxiliary system acts as a damped vibration absorber.

(a) Derive the equation of motion of the system plus absorber and then solve to determine what values of absorber mass \(m\), stiffness \(k_{2}\), and damping \(c\) must be selected in order to minimize the vibration amplitude of primary mass \(M\). Assume that \(m=0.25 M\) and \(F(t)=F_{0} \cos \omega t\).

(b) Suppose that we can accept a design where \(k_{2} M / k_{1} m\) is selected so that neither natural frequency of the combined system, \(\omega_{1}\) and \(\omega_{2}\), is closer than \(5 \%\) to the driving frequency \(\omega\). Devise the system that achieves this criterion.

Transcribed Image Text:

wwwww telle M F(1) F(t) m ellee M M telle k Figure 6.79: Original system and system with vibration: absorber.